In: Bulletin of the London Mathematical Society, 2020, vol. 52, no. 3, p. 472–488
A theorem of Dorronsoro from the 1980s quantifies the fact that real‐valued Sobolev functions on Euclidean spaces can be approximated by affine functions almost everywhere, and at all sufficiently small scales. We prove a variant of Dorronsoro's theorem in Heisenberg groups: functions in horizontal Sobolev spaces can be approximated by affine functions which are independent of the last...
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In: Advances in Mathematics, 2019, vol. 354, p. 106745
We study singular integral operators induced by 3-dimensional Calderón-Zygmund kernels in the Heisenberg group. We show that if such an operator is L2 bounded on vertical planes, with uniform constants, then it is also L2 bounded on all intrinsic graphs of compactly supported C1,α functions over vertical planes. In particular, the result applies to the operator R induced by the...
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In: Calculus of Variations and Partial Differential Equations, 2019, vol. 58, no. 2, p. 69
We show that a complete doubling metric space (X,d,μ) supports a weak 1-Poincaré inequality if and only if it admits a pencil of curves (PC) joining any pair of points s,t∈X . This notion was introduced by S. Semmes in the 90’s, and has been previously known to be a sufficient condition for the weak 1-Poincaré inequality. Our argument passes through the intermediate notion of a...
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In: Transactions of the American Mathematical Society, 2018, vol. 370, no. 7, p. 4909–4926
A family of planar curves is called a Moser family if it contains an isometric copy of every rectifiable curve in $ \mathbb{R}^{2}$ of length one. The classical ``worm problem'' of L. Moser from 1966 asks for the least area covered by the curves in any Moser family. In 1979, J. M. Marstrand proved that the answer is not zero: the union of curves in a Moser family always has area at least $ c$...
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